R E S E A R C H Open Access
Possible contribution of quantum-like
correlations to the placebo effect:
consequences on blind trials
91, Grande Rue, Sèvres, France
Background: Factors that participate in the biological changes associated with a
placebo are not completely understood. Natural evolution, mean regression,
concomitant procedures and other non specific effects are well-known factors that
contribute to the “placebo effect”. In this article, we suggest that quantum-like
correlations predicted by a probabilistic modeling could also play a role.
Results: An elementary experiment in biology or medicine comparing the biological
changes associated with two placebos is modeled. The originality of this modeling is
that experimenters, biological system and their interactions are described together
from the standpoint of a participant who is uninvolved in the measurement process.
Moreover, the small random probability fluctuations of a “real”experiment are also
taken into account. If both placebos are inert (with only different labels), common
sense suggests that the biological changes associated with the two placebos should
be comparable. However, the consequence of this modeling is the possibility for two
placebos to be associated with different outcomes due to the emergence of
Conclusion: The association of two placebos with different outcomes is
counterintuitive and this modeling could give a framework for some unexplained
observations where mere placebos are compared (in some alternative medicines for
example). This hypothesis can be tested in blind trials by comparing local vs. remote
assessment of correlations.
Keywords: Placebo effect, Quantum-like correlations, Experimenter effect,
Randomized clinical trials
Much has been written about the “placebo effect”and the purpose of this article is not
to make a review on this topic [1–6]. In itself the term “placebo effect”is curious and
paradoxical. Indeed, as underscored by Moerman and Jonas: “The one thing of which
we can be absolutely certain is that placebos do not cause placebo effects. Placebos are
inert and don’t cause anything”. For this reason, Ernst and Resch insisted to clarify
the definition of placebo by distinguishing “perceived placebo effect”and “true placebo
effect”. Perceived placebo effect is the outcome that is associated with the placebo
group in a trial; it includes natural evolution of the disease, mean regression, concomi-
tant procedures and other non specific effects. True placebo effect is the difference
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Beauvais Theoretical Biology and Medical Modelling (2017) 14:12
between perceived placebo effect and effect associated with no treatment. “No
treatment”groups are however infrequently performed and therefore there are often
some misunderstandings to define the scope of the “placebo effect”.
Since placebos are inert, the causes of the “true placebo effect”should be sought ra-
ther on the side of language and psychology. Thus, it has been shown that placebo ef-
fects can be caused by cognitive and emotional changes, expectation of symptom
changes or classical conditioning . Actual effects of placebo on brain and body have
been evidenced and there are neurobiological underpinnings for these effects [8, 9].
However, in most studies aimed to decipher the placebo effect, patients are at the
centre of the investigations and all explanations rest on them. In the present article, the
experimental design and the experimenters are also taken into consideration. There-
fore, the focus is moved from patients to investigators and in this case the placebo
effect –at least one of its components –is not much different than an experimenter’s
effect. A famous example of experimenter’s effect was evidenced in the experiments of
Rosenthal et al. where an experimenter obtained from his subjects the data he expected
or wanted to obtain . Outside of psychology, for example in cell biology or in physi-
ology, it is generally thought that such subtle influences could not be responsible for re-
sponse biases. In clinical trials, blind experiments are supposed to protect against any
outcome bias related to patient or physician; if such influences exist, they are distrib-
uted randomly in test and placebo groups. In the present article, it is suggested that
quantum-like correlations predicted by a probabilistic modeling could also contribute
to the “placebo effect”.
Design of a minimal experiment with two placebos
The purpose of a typical experiment in medicine or biology is to establish a relationship
between a “cause”(independent variable) and a biological “effect”. Placebos (or
“controls”in experimental biology) are included in the experiment in order to assess
the effects of variables other than the independent variable.
We define a biological “object”(biological model or patients in a clinical trial) with
two possible states: no biological change (or resting state, not different from back-
ground noise) and biological change (“activated”state). A biological change may be de-
fined by setting a cut-off value of a continuous variable. We symbolize no biological
change as “↓”and biological change as “↑”.
We assume that all samples that are tested are placebos and that the only difference
is their labels which are either Pcb
. Since samples are all inert and physically
identical, common sense suggests that the biological outcomes associated with the two
placebos should be comparable. Nevertheless, the aim of the modeling is to know
whether in some circumstances the state “↑”could be more frequently observed with
one of the two labels (no matter which one at this stage). Therefore, the null hypothesis
) of such an experiment is:
Prob (x∣y) is the conditional probability of xgiven y(or the probability of xunder
the condition y).
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 2 of 17
Figure 1 describes the two possible relationships between labels and biological out-
comes: either “direct”relationship (Pcb
associated with “↓”and Pcb
associated with “↑”and Pcb
associated with “↓”).
Note that this naming (“direct”or “reverse”) is arbitrary and does not prejudge results.
Of course, if a biological change is associated with the labels Pcb
same rate (i.e. no relationship), then the probabilities of direct and reverse relation-
ship are both equal to 1/2. By convention, we present calculations of probability mainly
for direct relationship (the sum of the probabilities of direct and reverse relationships is
equal to one).
Description of an experiment from an uninvolved standpoint
The originality of the present modeling is that observers, observed system and their in-
teractions are described together from an uninvolved standpoint. The formalism is in-
spired from the relational interpretation of quantum physics [11, 12] and quantum
Bayesianism (QBism) [13, 14].
We suppose that the experimental landscape is described by a participant who is
uninvolved in the experiment. Suppose,asdescribedinFig.2,anobserverOwho
measures a variable of a system S; this variable can take one of the two values
“left”and “right”after a measurement. For a participant Puninvolved in the meas-
urement process, a definite value has been obtained after the measurement of Sby
O(either “left”or “right”). Pknows that O has observed a defined value after
measurement, but Pdoes not know what O has observed. If Pfinally observes the
system S, he records a definite value and he agrees with Oon this value when P
and Ointeract. Interactions between observers are like measurements and they
allow establishing correlations.
In this last case, it is important to underscore that it is not correct to say that Pis
“forced”to observe what Oobserved before they interact. Indeed, one can imagine an-
other participant Qwho in turn describes S,Oand Pwithout interacting with them.
What Qcan say is that a correlation has been established between S,Oand P, but Q
Fig. 1 Relationships between placebos and biological system. There are two possible placebos (“0”and “1”)
and two possible states for the biological system: no change (“resting”state or background) which is noted
“↓”and biological change above background (“activated”state) which is noted “↑”. As a consequence, there
are two possible relationships defined as: direct relationship with “placebo 0”associated with “↓”and
“placebo”1 associated with “↑”; reverse relationship with “placebo 0”associated with “↑”and “placebo 1”
associated with “↓”
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 3 of 17
cannot say which result is observed. The only thing that an uninvolved participant can
do is to describe the form (correlations), but not the content (outcomes) of the informa-
tion available to the observers who interact with Sand with each other. Thus, the
consistency of any measurement is guaranteed.
Note that, strictly speaking, an uninvolved participant does not describe the “reality”
itself (made of “contents”), but he constructs a predictive tool (made of correlations
and probabilities) in order to know what to expect if he decides to interact with the
Probabilistic modeling of a minimal experiment with two placebos
An experiment is modeled from the standpoint of a participant Pwho is outside the la-
boratory, as described in the previous section. The participant Pdoes not interact with
the “objects”that he describes and he remains uninvolved in the evolution of the ex-
perimental situation. The role of Pis to describe the evolution of a team of interacting
experimenters with the knowledge of the initial conditions.
We consider a team composed of two experimenters named Oand O’who observe
the biological system S. We suppose an experimental situation where the probability
for each experimenter to observe a direct relationship (as defined in Fig. 1) is pand the
probability of a reverse relationship is q(with p+q= 1).
Each observer has his own probabilistic expectations and the uninvolved participant
Passigns the probability pto Oas the best estimate that Ocan make for the future ob-
servation of the direct relationship. The same probability pis assigned to O’independ-
ently of Osince the probabilistic expectations are specific to each observer.
Fig. 2 Description of an experiment from an uninvolved standpoint. The observer Omeasures the system
Swhereas the participant Premains uninvolved in the measurement (he does not interact with Oand S).
Pknows that Ohas observed a definite state of S, but he does not know which one. If Pfinally interacts
with Oand S, then Pand Oagree on the outcome of S. The reasoning can be continued with another
participant Qwho does not interact with S,Oand P. What Qcan say is that S,Oand Phave definite values
that are correlated. The only thing that an uninvolved participant can do is to describe the form, but not
the content of the information available to the observers who interact with Sand with each other. Thus,
the consistency of any measurement is guaranteed (GNU Free Documentation License)
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 4 of 17
Therefore, the participant Pconstructs a predictive tool for his own use where the ex-
perimenters have independent probabilistic expectations on the experimental outcome
and are in agreement when they compare their records. The future outcome that Oex-
pects to record (event A) and the future outcome that O’expects to record (event B)
are independent events in the probability space constructed by P(Fig. 3). This condi-
tion of independence is easily formalized since the probabilities of two independent
events Aand Bhave well-known mathematical properties:
Prob A∩BðÞ¼Prob AðÞProb BðÞ ð2Þ
Therefore, when Oand O’interact and agree on the result of the experiment (i.e. the
events of the set A∩B), the best estimate of the probability that Oand O’observe a
direct relationship is Prob (direct)=p×paccording to Eq. (2) since the probability to
record a direct relationship was estimated to be pfor Oand also pfor O’(Fig. 3). Simi-
larly, the best estimate of the probability that Oand O’observe a reverse relationship is
The intersubjective agreement discards some impossible situations such as Oobserves
a direct relationship while O’observes a reverse relationship. Since the sum of the prob-
abilities of all possible events is equal to one, Prob (direct)=p×pmust be renormalized.
For this purpose, p×pis divided by the sum of the probabilities of all possible outcomes
(grey areas in Fig. 3), namely direct relationship (p×p) and reverse relationship (q×q):
Fig. 3 Probabilistic space constructed by an uninvolved participant Pto predict the outcomes of the
experiments. A team of interacting experimenters Oand O’is described from the standpoint of an uninvolved
participant who knows the initial experimental conditions (Fig. 2). We suppose a probability equal to pfor the
event “direct relationship”and equal to qfor the event “reverse relationship”(p+q= 1). Each observer has his
own probabilistic expectations and Passigns the probability pto Oas the best estimate that Ocan make for
the future observation of a direct relationship; the same probability is assigned to O’independently of Osince
the probabilistic expectations are specific to each observer. White areas are unauthorized experimental
situations with incompatible outcomes after interaction of Oand O’(e.g. “direct”for Oand “reverse”for O’).
Therefore, the probability that the experimenters observe a direct relationship is calculated by dividing the
central gray area (“direct”for both observers) by the sum of the probabilities of possible outcomes (either
“direct”or “reverse”for both observers), namely all gray areas. Ω, probability space
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 5 of 17
By dividing both the numerator and the denominator by p
(taking into account that
p+q= 1), the only variable of the equation is p:
Prob directðÞ¼ 1
Eqs. (3), (4) and Fig. 3 are easily generalized to any number Nof observers who all
agree on the outcome:
Prob directðÞ¼ 1
In a “real”experiment, particularly in biology, random fluctuations occur and they
must be taken into account because, after each elementary fluctuation, a tiny bias is in-
troduced and Prob (direct) must be updated.
In the next lines we calculate the evolution of the probability for Oand O’to observe a
direct relationship from the standpoint of the participant P. First, we write that Prob
(direct) is equal to 1/2 in the absence of observers (N= 0 in Eq. (5)). As a consequence,
the initial value of Prob (direct)attimet
before the first fluctuation is equal to p
We then introduce ε
as successive elementary random fluctuations of Prob (direct)
that occur during successive elementary intervals of time (ε
are positive or negative
real random numbers such as ∣ε
∣< < 1). Note that an implicit consequence of the ran-
dom fluctuations of Prob (direct) is a non-null, but very small, probability to observe a
biological change (“↑”).
After the first fluctuation ε
, we easily calculate with Eq. (4) the updated probability
which is based on p
. The equation is then generalized for any probability p
based on previous probability p
and fluctuation ε
. We obtain a mathematical se-
quence which allows calculating the successive probabilities of a direct relationship:
Two placebos associated with different outcomes
Equation 6 allows calculating the successive states of a system constituted of a bio-
logical system and a team of interacting experimenters/observers committed in the es-
tablishment of a supposed relationship.
A computer calculation of this mathematical sequence is described in Fig. 4 after 100
successive random fluctuations ε
(with values around 10
) and with two observers
(N= 2). We observe that the initial situation is in fact metastable if fluctuations are
taken into account. Indeed, in all cases (i.e. whatever the series of values ε
), a dramatic
transition towards one of two stable positions is achieved:
Prob ðdirectÞ¼1=2ðmetastable positionÞ
Prob ðdirectÞ¼1or 0ðtwo possible stable positionsÞ
All samples of an experiment are thus engaged either in a direct relationship or in a
reverse relationship. Note that the probability of a biological change was very small
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 6 of 17
initially and, after the transition, a biological change is systematically associated with
the label Pcb
in stable position #1 or systematically associated with the label Pcb
stable position #2. The choice of one of the two stable positions is random. In both
cases, a relationship (direct or reverse) between labels and outcomes emerges.
However, the purpose of an experiment is to compare a “test”situation with a
“control”situation. Biological systems are therefore prepared in an asymmetrical state
with resting state (background noise) implicitly associated with a “control”situation.
The stability of the resting state (or basic state) is a condition for a proper assessment
of the samples (the experiment begins with the preparation of the biological system be-
fore samples are tested). In other terms, the state of the biological model at rest (before
each test) can be considered as associated with the label “control”.
We suppose, for example, that Pcb
is considered as a “control”by the experimenters.
Consequently the stable state #2 eliminates itself since Pcb
cannot be associated both
with change (when Pcb
samples are tested) and with no change (for the resting state).
Only the stable position #1 is a possible state:
Prob ðdirectÞ¼1=2ðmetastable positionÞ
Prob ðdirect Þ¼1ðstable positionÞ
A probability equal to one for the direct relationship means that the participant Pis
assured –if he finally interacts with the team of experimenters after the end of the
experiment –to observe a direct relationship between labels and biological outcomes.
Thanks to probability fluctuations, a biological change associated with each sample
label emerges from background noise.
Fig. 4 Calculation of the probability of a direct relationship. The evolution of the probability that a team
(composed of two members who interact) observes a direct relationship is described in this figure by taking
into account successive probability fluctuations. The probability defined in Fig. 3 is calculated according to
the mathematical sequence in the cartouche. Each successive probability p
of the sequence is calculated
by using p
and a random probability fluctuation is randomly obtained between −0.5 and +0.5 × 10
computer simulation shows that the initial state with a probability of 1/2 is in fact metastable and, after a
dramatic transition, one of two stable positions is achieved: either Prob (direct) = 1 or Prob (direct) = 0. With
N> 2 or with higher values of probability fluctuations, a transition is obtained after a lower number of
calculation steps (data not shown). Eight computer simulations are reported in this figure
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 7 of 17
Consequences of different types of blind designs on correlations
Until now we examined an experimental situation where the observers Oand O’
assessed themselves the rates of correlation between labels and biological outcomes
(open-label experiments). Nevertheless, the labels can be masked to the experi-
menters in order to reduce or eliminate any bias. After the outcomes have been
obtained in blind conditions for all samples, the labels of samples are unmasked.
In this article, we distinguish blind experiments with either local or remote assess-
ment of correlations.
For a local assessment of correlations with blind design, an automatic device or a
member of the team of experimenters keeps secret the labels of the samples until the
end of the experiment. In this case, the automatic device or the observer who is dedi-
cated to the blinding are also elements of the experiment because they interact with
the other observers and can be described (from the standpoint of P) with the same
modeling as open-label experiments.
A remote assessment of correlations with blind design is typically used in randomized
clinical trials (also named centralized blind design). The remote supervisor (a statisti-
cian for example) does not interact with the experimenters before all measurements are
completed. It is important to underscore that the remote supervisor should not be con-
fused with the uninvolved participant Pwho describes the experiment. Indeed, Pdoes
not interact and is not involved in the experiment. With a remote supervisor, the ex-
perimenters observe biological outcomes, but have no feedback on labels before the re-
mote experimenter is aware of the rate of success. As a consequence, Prob (direct)=
Prob (reverse); since Prob (direct) + Prob (reverse) = 1, then Prob (direct) = 1/2. In
Prob (direct) = 1 with local assessment of correlations;
Prob (direct) = 1/2 with remote assessment of correlations.
Figure 5 illustrates the consequences of the assessment of the correlations with a re-
mote assessment according to the modeling. In this case (blind experiment with an ex-
ternal supervisor), there is no statistical difference between the biological outcomes
associated with Pcb
in contrast with a local assessment (local blind design or
The experimental context is therefore crucial for establishing a relationship in
the modeling. With a local assessment, the experimenters observe labels and then
biological outcomes (open-label experiment) or observe biological outcomes and
then labels (local blind experiments). In contrast, with a remote supervisor, the ex-
perimenters observe biological outcomes, but have no feedback on labels. If a local
observer/experimenter is the first to assess the relationship, correlations emerge; if
a remote supervisor is the first to assess the relationship, correlations vanish (bio-
logical changes are nevertheless observed, but at random places). Of course, in all
cases, when participants met together, they agree on the conclusion (correlation or
no correlation). The order of the assessments (local first or remote first) is the key
element for the degree of correlation.
It is important to underscore that this difference between local and remote assess-
ment of correlations offers the opportunity to test the modeling.
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 8 of 17
Characterization of the role of the observers O and O’
The experimenters/observers Oand O’play a crucial role in the modeling and we
examine in this section how their involvement could be characterized and quantified.
As previously reported in Eq. (2), the joint probability of two independent events A
and Bis equal to the product of the separate probabilities of Aand B. This equation
can be generalized for two events Aand Baccording to their degree of independence:
Prob A∩BðÞ¼Prob AðÞProb BðÞþdwith 0 ≤d≤1ðÞ ð9Þ
The degree of independence increases when the value of ddecreases; the two events
are completely independent with d= 0. In other words, the correlation of the two
events increases when the value of dincreases. Eq. (3) can be easily modified if dis
taken into account (Fig. 6; see legend for calculation details):
p2þq2þ2dwith 0 ≤d≤pqðÞ ð10Þ
When the parameter dvaries from d=pq to d= 0, the experimental situation pro-
gressively shifts from a classical description to the present modeling (Fig. 6).
“Observing”an experiment requires a frame (what are we expecting?) and a feedback
(what did we record?). Equation (6) indicates that there is no transition of Prob (direct)
towards a stable position in the absence of observers (N= 0). We can draw the same
conclusion if the observers are physically present in the laboratory, but not focusing
Fig. 5 Comparison of local vs. remote assessment in an experiment with two placebos. In an experiment with
a local assessment (local blind design or open-label experiment), correlations between labels (Pcb
and states of the biological system (↓and ↑)emerge(band e) from the initial state (aand d). These correlations
vanish if the assessment of the experiment is made in a blind experiment with a remote supervisor (cand f). In
this latter case, the difference between the biological changes associated with Pcb
is not statistically
significant (NS) and the biological changes (“↑”) are randomly distributed among the two placebos. The
difference of results in local vs. remote assessments offers the opportunity to test the modeling
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 9 of 17
their attention on this specific experiment (they expect nothing about the experimental
system and they do not receive feedback). As a consequence, the parameter dcan be
considered as an evaluation of the attention of the team of experimenters to observe
the predefined relationship between labels and biological outcomes. When d= 0, the
observers are fully committed and for d=pq their attention is completely drawn away
from the experiment. For intermediate values, the team is more or less occupied with
these observations. Therefore, experimenters’qualities, such as attention, commitment
and persistence, appear to be necessary during the experiments for the emergence of
correlations between labels and biological outcomes.
We can go a step further by considering that the parameter dis also an assess-
ment of the capability of the experimenters to recognize (or not) the direct and re-
verse relationships per se, i.e. as new “objects”regardless their components, namely
the association of the biological outcomes with Pcb
. Indeed, in Eq. (4)
(i.e. with d= 0), it is implicit that the experimenters recognize the outcome per se
(i.e. in its “wholeness”or as such) as it would be the case for the outcome of a
dice roll or the position of a pointer on a measurement device. But suppose now a
team of experimenters Oand O’who are inexperienced and do not recognize the
predefined experimental relationship as a structured ensemble. The experimenters
identify the sub-events as separate elements without integrating them as a whole
(these sub-events are the association of the biological outcomes with Pcb
). Since we continue to adopt the standpoint of P, we use Eq. (4) to calculate
the evolution of the probability of each sub-event. Before the first fluctuation prob-
ability, the probabilities of the two sub-events are: Prob (direct |Pcb
)=1 and Prob
) and Prob (direct |Pcb
are already in stable positions. Therefore, by using Eq. (4) (see also Fig. 4), these condi-
tional probabilities are maintained in their respective stable positions with Prob (direct |
) that tends toward 1 and Prob (direct |Pcb
) that tends toward 0. The experimental
Fig. 6 From a classical description of the experimental situation to the present modeling. The experimental
situation depicted in Fig. 3 is generalized in this figure by using the parameter dwhich varies with the
degree of independence of the probabilistic expectations on the outcome assigned to Oand O’. The values
of the two areas with impossible situations (direct relationship for one observer and reverse relationship for
the other one) are calculated as: p–(p
+d)=p×(1 –p)–d=pq –d. For d= 0, correlations between labels
and biological outcomes emerge and, for d=pq, the probability of a direct relationship is equal to pas in
classical probability. Ω, probability space
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 10 of 17
results associated with these uncoupled sub-events can be gathered in order to calculate
Prob (direct) by using the law of total probability:
Prob directðÞ¼Prob Pcb0
As a consequence, because the relationship between labels and biological outcomes is
not recognized as a structured ensemble, there is no transition and Prob (direct) tends
These considerations are reminiscent from Gestalt psychology that states that human
mind spontaneously tends to perceive phenomena as structured ensembles (Gestalt)
and not as a simple addition of parts. For example, the well-known Necker cube is im-
mediately recognized as a 3D cube by a human observer and not as the simple addition
of lines drawn on a 2D sheet . We instantly “see”a cube in space because we have
learned to perceive these 2D drawings as 3D “objects”.
As for Necker cube, cognitive and learning processes are undoubtedly at work for the
passage from an “analytic”(d=pq)toa“structured”(or global) perspective (d= 0). In
the first situation (d=pq), the experimenters are spectators of the experimental landscape
that is perceived as a “collection of points”;inthesecondsituation(d= 0), they are actors
who interpret the experimental landscape that is perceived as a “form”. In this last case,
the experimenters concentrate their attention towards a “transcendent object”(namely, the
predefined relationship) without reference to the details that become indiscernible.
If cognitive and learning processes are involved in the emergence of quantum-like
correlations, different teams of experimenters with different training and experience
should report various degrees of correlations between labels and biological outcomes in
experiments comparing two placebos.
Emergence of a quantum-like logic
Only tools from classic probability are used in the modeling. Nevertheless, as demon-
strated in this section, there is an underlying quantum-like logic which is rooted in the
initial partition of placebos as Pcb
. Indeed, according to Fig. 1:
ðÞProb ↓ðÞþProb Pcb0
ðÞProb ↑ðÞþProb Pcb1
When the stable position #1 is achieved, Prob (Pcb
) = Prob (↓) and Prob (Pcb
Prob (↑) (see Fig. 5b); when the stable position #2 is achieved, Prob (Pcb
) = Prob (↑)
and Prob (Pcb
) = Prob (↓). Therefore in both cases:
This equation is equivalent to:
Then, we define aand bsuch as Prob (Pcb
(or b.b). These definitions correspond to the stable position #1 (for the stable pos-
ition #2, b
must be taken equal to –b×−b):
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 11 of 17
As can be seen in Fig. 7, the left-hand side of Eq. (17) is the sum of Prob (direct) plus
Prob (reverse) without a remote supervisor and the right-hand side is the sum of Prob
(direct) plus Prob (reverse) with a remote supervisor. The terms aand bare thus
probability amplitudes and their squaring allows calculating the corresponding
Therefore, the probability of a direct relationship without a remote supervisor is cal-
culated by doing the sum of the probability amplitudes of the two paths that lead to a
direct relationship and then by squaring this sum. With a remote supervisor, the prob-
ability of a direct relationship is calculated by squaring the probability amplitude of
each path that leads to a direct relationship and then by making the sum of the prob-
abilities of the two paths (Fig. 7).
The relationship between labels and biological outcomes in the modeling has the
same logic as single-photon self-interferences in Young’s double-slit experiment where
photons behave either as particles when paths are detected or as waves when paths are
not detected. In Fig. 7 that sketches an elementary experiment, quantum-like correla-
tions are observed when “paths”(i.e. labels) are undistinguishable (from an outside
standpoint) and correlations vanish when they are distinguishable for a remote super-
visor. In this last case, each label is forced to adopt a defined “pathway”.
The emergence of quantum-like correlations is the consequence of the initial as-
sumptions, namely the independent probabilistic expectations and the intersubjective
agreement. The concomitant consideration of these two assumptions implies that the
outcome of an experiment does not pre-exist to the interaction of Oand O’from the
standpoint of P. This is a characteristic of quantum measurements and, in the language
of quantum mechanics, the “state”of Oconcerning his identification of the outcome
Fig. 7 Probability of a direct relationship without or with a remote supervisor. The quantum-like probability
of a direct relationship is calculated as the square of the sum of the probability amplitudes of the different
possible “paths”. With a remote supervisor, classical probability applies and the probability of a direct
relationship is calculated as the sum of squares of the probability amplitudes of the “paths”. Therefore, the
probabilities of a direct relationship are different without or with a remote supervisor
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 12 of 17
would be said “superposed”before interacting with O’(and vice versa). The intersub-
jective agreement plays a similar role as a conservative law in physics and Oand O’
would be said “entangled”after their interaction.
Importance of an uninvolved standpoint
The uninvolved standpoint of the participant Pis central in the construction of the
modeling. Indeed, from the standpoint of O, if he observes a direct relationship or a re-
verse relationship, then he can hold for sure that O’will tell him that he observes the
same event. As a consequence the probability that Oand O’observe a direct relation-
ship is pin this case as stated by classical probability and not p×p(before
renormalization) from the standpoint of P. The standpoints of Pand O-O’coincide in
situations where these two equations are verified:
These two equations are equivalent to (2p–1) (p–1) = 0 and (2q–1) (q–1) = 0, re-
spectively. Therefore, there are only three possible values for p: 1/2, 1 or 0. These
values are the probabilities associated with initial position, stable positions #1 and #2,
respectively. Only the outside standpoint of Pwho is not involved in the observation of
the experiment allows describing the transition of Prob (direct) from 1/2 to 1 (or 0) as
a consequence of the emergence of quantum-like “interferences”(i.e. the cross-terms
with probability amplitudes equal to band -b in Fig. 7).
The differences between the standpoints of O-O’and Pare the consequence of the
demonstration of Breuer about the impossibility of a complete self-measurement .
According to this demonstration, a measurement apparatus (or an observer) is unable
to distinguish all the states of a system in which it is contained (whether this system is
classical or quantum mechanical does not matter). Only a second external apparatus
(P) that observes both the first apparatus (O) and the system (S) is able to account all
correlations between Oand S.
Optimized placebos in clinical trials
Without any doubt, the success of many complementary or alternative medicines rests
on placebo effect. Thus, most authors consider homeopathy as a perfect illustration of
the enforcement of the placebo effect in medicine. Moreover, homeopathic medicines
could be considered as “super placebos”(or optimized placebos) since even practi-
tioners think that they prescribe “true”medicines despite the absence of active mole-
cules. Indeed, the manufacturing process of a majority of homeopathic medicines
eliminates the initial active molecules by serially diluting them well beyond the limit set
by Avogadro’s number. In other words, there are zero active molecules in these highly
diluted samples. Even if tiny traces of the initial molecules would be present (due to
contamination or imperfect diluting process), it remains to demonstrate how they
could nevertheless have an effect contradicting the law of mass action.
Since no classical pharmacological action can be assigned to high dilutions, it has
been suggested that modifications of water structure during the dilution process could
account for the alleged effects. Until now, no convincing evidence has been reported
indicating that modifications of water structure specific of the initial molecules are able
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 13 of 17
to induce specific biologic changes. Moreover, homeopathy medicines available in phar-
macies are sugar granules that have been impregnated with high dilutions and then
dried. Therefore, until there is evidence to the contrary, the most reasonable scientific
attitude is to consider homeopathy medicines and high dilutions as plain placebos.
Interestingly, the gold standard for drug evaluation, namely blind randomized clinical
trial, appears to be an obstacle in studies aimed to establish the efficacy of homeopathy
medicines. Thus, the study of Shang et al. compared homeopathy trials and matched
conventional-medicine trials [18, 19]. The authors concluded that homeopathy medi-
cines were comparable to placebos. Indeed, in contrast with conventional medicines,
double-blind design was associated with a strong decrease of the probability of success
when compared with open-label design. Although this study has been heavily criticized
by proponents of homeopathy, most of them nevertheless acknowledge that blind ran-
domized clinical trials are not adequate for assessing homeopathy medicines [20, 21]. A
randomized clinical trial by Brien et al. in patients with rheumatoid arthritis suggested
that homeopathy consultations, but not homeopathy medicines, were associated with a
clinical benefit thus reinforcing the idea of a placebo effect .
In 2013, I proposed a slight modification of trial design in order to increase the
chance to observe a difference between outcomes in double-blind placebo-controlled
randomized trials of homeopathy medicines. This suggestion was not an encourage-
ment for the practice of homeopathy, but an attempt to understand the persisting suc-
cess of this alternative medicine in the absence of a rational basis. Based on the
hypothesis that quantum-like correlations were responsible for “successful”open-label
homeopathy clinical trials, it was proposed to replace the centralized assessment of effi-
cacy in blind trials (generally done by statisticians) with a local assessment (by physi-
cians) . Thieves et al. recently challenged this hypothesis and reported experiments
in a plant model (wheat germination) that compared a homeopathy medicine and a pla-
cebo both in local and centralized blind designs . The results were in favor of the
initial hypothesis since a significant difference of plant growth was observed between
homeopathy medicine and placebo with local assessment while there was no significant
difference with centralized assessment. The interaction test for local vs. centralized
blind designs was statistically significant (p = 0.003). If we consider all samples (includ-
ing homeopathy medicine) as plain placebos that differ only by their labels, these re-
sults are in favor of the present hypothetical modeling. These results should be also an
encouragement for physicians to implement the same local blind design in clinical trials
comparing a placebo with homeopathy medicine (i.e. a second placebo) in order to test
in vivo the hypothesis of quantum-like correlations as depicted in Fig. 5.
It is generally thought that the macroscopic world escapes to the consequences of
quantum physics due to the decoherence process. As a consequence, biological systems
are considered to behave only classically. Nevertheless, some phenomena such as
photosynthetic light harvesting or avian magnetoreception have been recently sug-
gested to be the consequence of quantum phenomena . Asano et al. evidenced
quantum-like probabilistic behavior in Escherichia coli lactose-glucose metabolism .
In experimental psychology, some processes of cognition appear to obey to nonclassical
logic . Thus, the purpose of the new field named "quantum cognition”is to describe
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 14 of 17
cognitive processes such as reasoning, decision making, judgment, language, memory
or perception with mathematical quantum tools [28–31]. Moreover, Aerts described
some experimental situations in physics where macroscopic devices could exhibit a
quantum-like behavior . Interestingly, Aerts showed that quantum probabilities
could be introduced as the consequence of a lack of knowledge about fluctuations dur-
ing the interaction between a measuring device and the object to be measured .
Most authors that use quantum probability outside the field of physics do not consider
that the systems they describe are really quantum. Tools of quantum probability are
simply used to describe results that until then were considered paradoxical . In-
deed, quantum physics is not only a new mechanics but also a new probability theory.
An extension of classical probability with some mathematical tools borrowed to
quantum probability (e.g. superposition, entanglement, interferences) appears to be
fruitful in these different domains. With the present hypothetical modeling, it is pro-
posed that quantum-like correlations could be a component of the placebo effect.
A central question is the generalisability of the proposed modeling to other experi-
mental situations. Indeed, one could argue that bets on a coin toss could be also de-
scribed by the same modeling by replacing labels with bets and biological system with
coin toss. The answer is in Eq. (6) that supposes first that the system Shas an internal
structure submitted to small random fluctuations (thermal fluctuations for example)
and second that each p
value is strongly dependent on p
value. In other words,
are correlated with probabilities p
. This last characteristic is named
temporal autocorrelation and is a feature of phenomena with slow random fluctuations
such as systems submitted to Brownian motion or biological systems. Of course, an-
other implicit condition is the absence of physical obstacles that would block the transi-
tion of Prob (direct). Therefore, for systems based on a phenomenon not submitted to
internal fluctuations (radioactive decay) or “rigid”systems with sufficient mechanical
inertia to be not influenced (coin flipping or dice rolling), εis equal to zero and no
transition is possible. For experimental systems submitted to internal fluctuations, but
with successive states that are not autocorrelated due to strong restoring forces
(“elastic”systems), a transition as described in Fig. 4 is not possible (only random fluc-
tuations around 1/2 are observed). An example of such a system is a beam splitter that
randomly transmits or reflects a photon and vibrates around a fixed point. Systems
based on phenomena with large random fluctuations (electronic noise for example) are
The possibility to be experimentally tested is the hallmark of a scientific theory.
The proposed modeling predicts that quantum-like correlations vanish when they
are assessed by a remote supervisor. Only local assessments allow quantum-like
“interferences”with correlation of “expected”and observed outcomes. It is import-
ant to emphasize that this modeling does not describe a causal relationship be-
tween mental states (e.g. intention) and physical states. Indeed, only quantum-like
correlations are allowed and there is no way to transmit messages, instructions or
orders from a laboratory to another one by using a series of coded samples.
Walach has extensively studied the relationship between homeopathy and notions
from quantum logic such as complementarity and entanglement by using a “generalized
quantum theory”[33, 34]. Of interest, this author insisted that homeopathy medicines
and their associated clinical outcomes could not be treated causally (as it the case in
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 15 of 17
blind randomized clinical trials), otherwise mismatches between outcomes occurred
. The present modeling with two placebo, which are differently labeled, leads to the
same conclusion. Moreover, it is not excluded that quantum-like correlations could
emerge in clinical trials for conventional drugs and add to classical causal relationship.
Some authors reported clinical trials where placebos associated with different labels
or therapeutical rituals could lead to different outcomes [36, 37]. Only psychological
mechanisms were supposed to be the cause of the different outcomes. Nevertheless, it
would be interesting to evaluate a possible involvement of quantum-like correlations in
such experiments aimed at investigating the placebo effect.
The potential existence of quantum-like correlations in the context of the experi-
menter effect could be also an element interesting to explore in the current debate
about low reproducibility in life sciences . Indeed, differences among experimenters’
teams are expected for the establishment of quantum correlations according to the
modeling. As a matter of fact, trials in biology, medicine or psychology could benefit
from an extended theory of probability that permits interferences between probabilities
(more exactly between probability amplitudes).
The hypothetical modeling proposed in this article suggests that two placebos with dif-
ferent labels can be associated with different outcomes even in blind trials. Such a
counterintuitive conclusion is the consequence of a probabilistic modeling that autho-
rizes quantum-like interferences. This modeling could give a framework for some unex-
plained observations where mere placebos are compared (in some alternative
medicines for example) and could be tested in blind trials by comparing local vs. re-
mote assessment of correlations.
Availability of data and material
Not applicable (unique author)
The author declares that he has no competing interests
Consent for publication
Ethics approval and consent to participate
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 15 February 2017 Accepted: 25 May 2017
1. Moerman DE, Jonas WB. Deconstructing the placebo effect and finding the meaning response. Ann Intern Med.
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 16 of 17
2. Price DD, Finniss DG, Benedetti F. A comprehensive review of the placebo effect: recent advances and current
thought. Annu Rev Psychol. 2008;59:565–90.
3. Meissner K, Kohls N, Colloca L. Introduction to placebo effects in medicine: mechanisms and clinical implications.
Philos Trans R Soc Lond B Biol Sci. 2011;366:1783–9.
4. Geers AL, Miller FG. Understanding and translating the knowledge about placebo effects: the contribution of
psychology. Curr Opin Psychiatry. 2014;27:326–31.
5. Coste J, Montel S. Placebo-related effects: a meta-narrative review of conceptualization, mechanisms and their
relevance in rheumatology. Rheumatology (Oxford). 2017;56:334–43.
6. Shapiro AK. A contribution to a history of the placebo effect. Syst Res Behav Sci. 1960;5:109–35.
7. Ernst E, Resch KL. Concept of true and perceived placebo effects. BMJ. 1995;311:551–3.
8. Lieberman MD, Jarcho JM, Berman S, Naliboff BD, Suyenobu BY, Mandelkern M, Mayer EA. The neural correlates of
placebo effects: a disruption account. Neuroimage. 2004;22:447–55.
9. Price DD, Craggs J, Verne GN, Perlstein WM, Robinson ME. Placebo analgesia is accompanied by large reductions
in pain-related brain activity in irritable bowel syndrome patients. Pain. 2007;127:63–72.
10. Rosenthal R, Fode KL. The effect of experimenter bias on the performance of the albino rat. Behav Sci. 1963;8:183–9.
11. Rovelli C. Relational quantum mechanics. Int J Theor Phys. 1996;35:1637–78.
12. Smerlak M, Rovelli C. Relational EPR. Found Phys. 2007;37:427–45.
13. Fuchs CA: QBism, the perimeter of quantum Bayesianism. Arxiv preprint 2010:arXiv:1003.5209.
14. Fuchs CA, Mermin ND, Schack R: An introduction to QBism with an application to the locality of quantum
mechanics. Arxiv preprint 2013:arXiv:1311.5253.
15. Bitbol M. The quantum structure of knowledge. Axiomathes. 2011;21:357–71.
16. Breuer T. The impossibility of accurate state self-measurements. Philos Sci. 1995;62:197–214.
17. Laudisa F, Rovelli C: “Relational Quantum Mechanics”, The Stanford Encyclopedia of Philosophy (Summer 2013
Edition), Zalta EN (ed.). Available at https://plato.stanford.edu/entries/qm-relational/.
18. Shang A, Huwiler-Muntener K, Nartey L, Juni P, Dorig S, Sterne JA, Pewsner D, Egger M. Are the clinical effects of
homoeopathy placebo effects? comparative study of placebo-controlled trials of homoeopathy and allopathy.
19. Vandenbroucke JP. Homoeopathy and “the growth of truth”. Lancet. 2005;366:691–2.
20. Weatherley-Jones E, Thompson EA, Thomas KJ. The placebo-controlled trial as a test of complementary and
alternative medicine: observations from research experience of individualised homeopathic treatment.
21. Milgrom LR. Gold standards, golden calves, and random reproducibility: why homeopaths at last have something
to smile about. J Altern Complement Med. 2009;15:205–7.
22. Brien S, Lachance L, Prescott P, McDermott C, Lewith G. Homeopathy has clinical benefits in rheumatoid arthritis
patients that are attributable to the consultation process but not the homeopathic remedy: a randomized
controlled clinical trial. Rheumatology (Oxford). 2011;50:1070–82.
23. Beauvais F. A quantum-like model of homeopathy clinical trials: importance of in situ randomization and
unblinding. Homeopathy. 2013;102:106–13.
24. Thieves K, Gleiss A, Kratky KW, Frass M. First evidence of Beauvais’hypothesis in a plant model. Homeopathy.
25. Lambert N, Chen Y-N, Cheng Y-C, Li C-M, Chen G-Y, Nori F. Quantum biology. Nat Phys. 2013;9:10–8.
26. Asano M, Khrennikov A, Ohya M, Tanaka Y, Yamato I. Three-body system metaphor for the two-slit experiment
and Escherichia coli lactose-glucose metabolism. Philos Trans A Math Phys Eng Sci. 2016;374(2068).
27. Busemeyer J, Bruza P. Quantum models of cognition and decision. New York: Cambridge University Press. 2012.
28. Busemeyer JR, Wang Z, Khrennikov A, Basieva I. Applying quantum principles to psychology. Phys Scr.
29. Asano M, Hashimoto T, Khrennikov A, Ohya M, Tanaka Y. Violation of contextual generalization of the Leggett–
Garg inequality for recognition of ambiguous figures. Phys Scr. 2014;T163:014006.
30. Tressoldi PE, Maier MA, Buechner VL, Khrennikov A. A macroscopic violation of no-signaling in time inequalities?
How to test temporal entanglement with behavioral observables. Front Psychol. 2015;6:1061.
31. Khrennikov A, Basieva I. Possibility to agree on disagree from quantum information and decision making. J Math
32. Aerts D. Quantum structures due to fluctuations of the measurement situations. Int J Theor Phys. 1993;32:2207–20.
33. Atmanspacher H, Römer H, Walach H. Weak quantum theory: complementarity and entanglement in physics and
beyond. Found Phys. 2002;32:379–406.
34. Walach H, von Stillfried N. Generalised quantum theory - basic idea and general intuition: a background story and
overview. Axiomathes. 2011;21:185–209.
35. Walach H. Entangled–and tied in knots! practical consequences of an entanglement model for homeopathic
research and practice. Homeopathy. 2005;94:96–9.
36. Whalley B, Hyland ME, Kirsch I. Consistency of the placebo effect. J Psychosom Res. 2008;64:537–41.
37. Hyland ME, Whalley B. Motivational concordance: an important mechanism in self-help therapeutic rituals
involving inert (placebo) substances. J Psychosom Res. 2008;65:405–13.
38. Reality check on reproducibility. Nature. 2016;533(7604):437.
Beauvais Theoretical Biology and Medical Modelling (2017) 14:12 Page 17 of 17
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